Question 6
Triangle $ABC$ has incenter $I$, incircle $\gamma$ and circumcircle $\Gamma$. Let $M,N,P$ be the midpoints of $BC,CA,AB$, and $E,F$ the tangency points of $\gamma$ with $CA,AB$. Set $U=EF\cap MN$, $V=EF\cap MP$ using the full lines, and let $X$ be the midpoint of arc $BAC$ on $\Gamma$. (a) Prove that $I$ lies on ray $CV$. (b) Prove that line $XI$ bisects $UV$.
Proposed by Titu Andreescu and Cosmin Pohoata (credited in the supplied notes).
Proposed by Titu Andreescu and Cosmin Pohoata (credited in the supplied notes).